R70-38 The Time Required for Group Multiplication

Applying some simple, easily understood principles, Spira, in extending some earlier work of Winograd, points the way to a powerful theory of computation complexity. Spira considers a (d, r) combinational network which is an interconnection of r-input, single-output modules, with each input-output line carrying a value from the set {0, 1, ⋯ , d -1}. A finite function φ: X1 × X2 × ⋯ Xn→Y is to be computed, but it is assumed that before the inputs are inserted into the network, each input can be individually (and arbitrarily) transformed by a set of maps gj: Xj→Ij. It is also assumed that there is a 1-1 output map h: Y→Oc, so in practice the (d, r) network will have as input [g1(x), ⋯, gn(xn)] and as output h(φ(x1, ⋯, xn)). The problem is to bound the number of levels required of the network. Given a φ for a particular output mapping, it is not difficult to specify a lower bound on the number of levels required, by identifying for each output line the number of different values of input variables which yield a different output value. The minimum number of levels required for each output line is then evaluated by noting that an output at level z can depend on at most r´ input lines whence the output line requiring the most levels provides the bound.